Equation of the Month

Equation of the Month

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Tuesday, February 28, 2012

What it means
It states that the correlation between the densities of two separate, conspecific, populations (ρp), is equal to the correlation between their respective environments (ρe). This offers a simple explanation to the often found synchrony of the dynamics of spatially separated populations. Populations densities vary in synchrony because local environments are correlated across space. ‘Environment’ is here interpreted in the broad sense, it can be abiotic (e.g. weather) or biotic (e.g. predation pressure)).

Where does it come from
Moran (1953), in a paper on the dynamics of the highly synchronized Canadian lynx populations, stated the theorem without really proving it (”It can easily be shown mathematically that...”). It is based on a set of simplifying assumptions:

i) Each local population is driven by linear, stochastic dynamics. A simple example is a first order auto-regressive process (AR(1)): xt = axt-1 + εt, where a is a constant, xt is (possibly log-transformed) population density at time t (minus its long term mean) and εt is the local environment at time t.

ii) All local populations are driven by exactly the same dynamic equation.

iii) All environmental fluctuations are either temporally uncorrelated ('white noise') or share the same temporal structure (they could be linear, auto-regressive processes themselves).

iv) There is no dispersal between populations

The theorem received little attention until Royama (1984, 1992) brought it up and coined its name.

Applicability and importance
More realistic assumptions (non-linear, unequal dynamics) lead to relatively lower population synchrony, compared to Moran's prediction (e.g. Ranta et al 2006). For natural populations one can thus not assume that the Moran effect is as strong as in the ideal case. Its true power lies in its generality. Any structured, linear, model yields the same result. It is thus applicable, at least approximately, to in principle all natural populations, offering an always-present explanation to synchrony. As an example, many cyclic populations are highly synchronized. It therefore tempting to look for a single mechanism causing both the cycles and the synchrony. The Moran effect readily explains the synchrony. Other explanations (such as predator-prey interactions) can be sought for the cyclicity (Royama 1992).

The major alternative explanations to population synchrony that have been put forward are dispersal between populations and nomadic predators. Especially the role of dispersal has been analysed in some detail, showing a strong dependence on the character of the local dynamics. In any case, the Moran effect is always present, it can never be ignored.

From a conservation point of view, population synchrony decreases the viability of spatially structured populations. In short, it increases the probability that several local population go extinct simultaneously. This is in contrast to the mixed blessing of dispersal, which increases synchrony but at the same time makes possible recolonization of empty habitat patches.

Jörgen Ripa

Further reading
Moran, P. A. P. 1953. The statistical analysis of the Canadian lynx cycle. II. Synchronization and meteorology. Australian Journal of Zoology 1: 291-298.